Most popular preconditioners are implicit, that is, to apply them
one has to solve a system. One might say that they compute an approximation
to the coefficient matrix that is easier to solve with than the matrix
itself.

The approximate inverse class of preconditioners
is different in that they compute explicitly an approximation
to the inverse. Hence the application is an explicit matrix-vector
product operation, and therefore trivially parallel.

The method in the SPAI code is based on ideas from [5]:
the minimisation problem or is solved,
with the sparsity
pattern of M predetermined or adaptively determined.
This minimisation problem turns out to reduce to independent subproblems
for the rows or columns of M, and is therefore executable in parallel.

An other advantage of this method is that it is not subject to breakdown
the way factorisation based methods are.